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Ömür Uğur
2025-09-17 13:28:33 +03:00
parent 61b567ec19
commit 094778f67e
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tutorials/tutorial3_orthogonalization.ipynb
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@@ -94,7 +94,7 @@
"q_1 = \\frac{a_1}{\\|a_1\\|}\n",
"$$\n",
"$$\n",
"q_k = \\frac{a_k - \\sum_{j=1}^{k-1} (q_j \\cdot a_k) q_j}{\\left\\|a_k - \\sum_{j=1}^{k-1} (q_j \\cdot a_k) q_j\\right\\|}\n",
"q_k = \\frac{a_k - \\sum_{j=1}^{k-1} (q_j \\cdot a_k) q_j}{\\left\\|a_k - \\sum_{j=1}^{k-1} (q_j \\cdot a_k) q_j\\right\\|}, \\qquad k = 2, \\ldots, n\n",
"$$\n",
"\n",
"Matrix form:\n",
@@ -236,17 +236,17 @@
"\n",
"We want to solve\n",
"\n",
"$$ \\min_x \\|Ax - b\\|_2. $$\n",
"$$ \\min_x \\Vert Ax - b \\Vert_2^2. $$\n",
"\n",
"If $A = QR$, then\n",
"\n",
"$$ \\min_x \\|Ax - b\\|_2 = \\min_x \\|QRx - b\\|_2. $$\n",
"$$ \\min_x \\Vert Ax - b \\Vert_2^2 = \\min_x \\Vert QRx - b \\Vert_2^2. $$\n",
"\n",
"Since $Q$ has orthonormal columns:\n",
"Since $Q$ has orthonormal columns, and the normal equations boils down to\n",
"\n",
"$$ R x = Q^T b. $$\n",
"$$ R x = Q^T b, $$\n",
"\n",
"So we can solve using back-substitution.\n"
"we can therefore solve for $x$ by using back-substitution.\n"
]
},
{